Functions of several variables

Functions of several variables

Lessons

Notes:

2 Variable Functions
Functions with two variables are written in 2 forms:

z=f(x,y)z = f(x,y)
F(x,y,z)=0F(x,y,z) = 0


Domain & Range
The domain and range of a function f(x,y)f(x,y) is as follows:

D={(x,y)R2:f(x,y)} D = \{ (x,y) \in \mathbb{R}^2 : f(x,y) \}
R={zR:f(x,y)=z}R = \{ z \in \mathbb{R} : f(x,y) = z \}


Trace

A trace of a surface is a set of intersecting points which is created by a surface intersecting one of the following planes: xyxy-plane, xzxz-plane, yzyz-plane.

Trace of the xyxy-plane happens at z=0z=0.
Trace of the xzxz-plane happens at y=0y=0.
Trace of the yzyz-plane happens at x=0x=0.

Level Curve
A level curve is a curve that is on the xyxy-plane with a specific value z=z0z=z_0.

  • Introduction
    Functions of Several Variables Overview:
    a)
    2 Variable Functions
    • How to represent them
    • What they look like Visually

    b)
    Domain & Range
    • Domain \to looks at xx and yy
    • Range \to looks at zz
    • Examples of finding Domain & Range

    c)
    Traces
    • Traces \to intersection of surface and planes
    • What it looks like visually
    • Example of finding the Trace

    d)
    Level Curves
    • Curves on the xyxy-plane with a specific zz value
    • Examples of finding level curves


  • 1.
    Finding Domain & Range
    Find the domain and range of:

    f(x,y)=1x2+y24 f(x,y) = \frac{1}{\sqrt{x^2 + y^2 - 4}}


  • 2.
    Find the domain and range of:

    f(x,y)=xx2+y2e1y2f(x,y) = \sqrt{x} \sqrt{x^2+y^2} e^{1-y^2}


  • 3.
    Find the domain and range of:

    f(x,y)=ln(x21)f(x,y) = ln(x^2 - 1)


  • 4.
    Finding the Traces
    Find and graph the trace of the xzxz plane of

    f(x,y)=xx+y2f(x,y) = \sqrt{x}\sqrt{x+y^2}


  • 5.
    Find and graph the trace of the yzyz plane of

    5x+2y+z=35x + 2y + z = 3


  • 6.
    Find the level curve of f(x,y)=x2+y2f(x,y) = \sqrt{x^2 + y^2} at z0=1z_0 = 1 and z1=2z_1 = 2

  • 7.
    Find and graph the level curve of 2z+4y2x=02z + 4y^2 - x = 0 at z0=0z_0 = 0.